A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Why are Optional Stochastic Processes Important?

I understand to some degree why adapted processes, progressive processes, and predictable processes are important. But why do we care about optional processes? What is significant about the optional ...
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Finding the mean given the probability

I'm doing some work on branching processes and would like to know where the process becomes extinct. If $X$ is the number of offspring of an individual, then the process goes extinct when ...
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Expected Square Distance from Origin of Random Walk in $\mathbb{Z}^2$

I'm trying to find the expected value of the squared distance from the origin of a simple symmetric random walk in $\mathbb{Z}^2$ at time $n$. So far, I have calculated that if $(X,Y)$ is the ...
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System of SDEs and independence

I am recently reading a paper that seems to claim the following fact without justification: $Y^1_t, \ldots, Y^n_t$ are stochastic processes defined on $\mathbb{R}$. Let $b: \mathbb{R}^2 ...
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How do linear operators acting on paths of Gaussian processes influence the covariance function?

It is well-known that applying a linear transformation $A$ on an $n$-dimensional centered Gaussian distribution with covariance matrix $\Sigma$ results in another centered Gaussian distribution with ...
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canonical process satisfies SDE too

Let \begin{align}\label{11} dX_t=\sigma(X_t)\, dW_t+b(X_t)\,dt,\ \ \ X_0=x. \end{align} where $\sigma$ and $b$ are both Lipschitz. According to Picard-iteration-theorem, there exists a solution. Let ...
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Why Are Semimartingales the Largest Possible Class of Stochastic Integrators?

I am trying to understand why semimartingales are the most general possible class of stochastic integrators. (I was hoping that this question would give me my answer, but it didn't.) I thought at ...
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28 views

What is the probability that a given $ n $ event trains match the beginning of a Poisson process?

Here is my question with which I'm confusing myself: Assume that some event times $ \{\tau_i\}_{i \in \mathbb{N}} $ are a point process with rate $ \mu $ such that number of events that occurred ...
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Functions of a random walk and martingales

Let $\xi_1,\xi_2,\ldots$ be a sequence of iid random variables, such that $$\mathbb{P}(\xi_i=1)=p\ne \frac{1}{2},\,\mathbb{P}(\xi_i=-1)=q=1-p.$$ Consider the corresponding random walk ...
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Convergence of a sequence over supremum

Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$. For $n\in \mathbb{N}$ the ...
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Uncorrelated but not independent uniform distribution

Let $X = (X_1, X_2)$ be uniform distributed on $\{(-1,0), (1,0), (0,-1), (0,1)\}$. First of all I want to show that $X_1$ and $X_2$ are uncorrelated but not independent. Secondly I thought about ...
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Do Optional and Progressive Processes Have Counterparts in Discrete Time?

We know that predictable $\implies$ optional $\implies$ progressively measurable. Source Predictable processes have obvious/simple counterparts in discrete time. Do optional processes and ...
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Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv ...
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Hard Question in Stochastic processes - variance Martingales

I got some hard challenge to solve and I am looking for a small clue/help. My question goes like this: 10 Englishmen are trying to leave a pub in a rainy weather. They do it in the following ...
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$2D$ random walk stopping time

A $2D$ random walk starts at $(X_0, Y_0) = (k, k)$ where $k>0$ is an integer. At each step $(X_{n+1}, Y_{n+1}) = (X_{n}-1, Y_{n})$ or $(X_{n+1}, Y_{n+1}) = (X_{n}, Y_{n}-1)$ with the same ...
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What is the difference between an adapted process and a predictable process?

As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is. However, since I don't have any idea ...
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Conditional Probability in Poisson Process

Suppose that $N(t), t\ge 0$ is a Poisson process such the $E[N(9)]=6$. (i) Find the mean and variance of $N(8)$. (ii) Find $P(N(2)\le3)$ (iii) Find $P(N(4)\le5|N(2)\le3)$ - How do I solve this? ...
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Indicator Functions - Can someone check my working?

This is a very easy question but since some of my codes aren't coming out properly I thought I should check my theory to see if everything's okay. Say we have two values $K_{1}$ and $K_{2}$ and that ...
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1answer
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Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
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Esscher-Transform/ Levy-Process: Measure induced by trajectory

For a Levy-process $X_t$ w.r.t. to a measure P we define $\Theta$ as the set, for which $E[exp(\theta X_t)]$ is defined and finite. Note $\Theta$ is independent of $X_t$. Define ...
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To test whether a process is a Martingale (Stochastic calculus)?

If $W_t$ is a standard Brownian motion, I was trying to prove $Y_t = \exp (\int_{0}^{t} s\cdot dW_s)$ is a martingale ! First I started finding $dY_t$ using Ito formula. But I am confused how to ...
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Integral equation with a random variable [on hold]

Suppose we have an integtal of this kind: $$\displaystyle Q(\tau)=\int_0^\tau dt K(w(t)+\nu(t))$$ where $\nu(t)$ is a gaussian noise $\nu(t)=\mathcal{N}(0,\sigma)$ Because $w(t)$ is a known function, ...
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How to prove that the stochastic integral process is gaussian?

I would like to prove that for a $C^1$-function f and a Wiener process W, the integral process defined by $$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$ Is a centered gaussian ...
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Residual life distribution for renewal process after time T

Suppose we have a renewal process with inter-arrival times $\boldsymbol X=\{X_1, X_2, ...\}$, where $X_i$ are i.i.d variables. Assume that the CDF and PDF for $X_i$ is $F(x)$ and $f(x)$. 1) Let $A_t$ ...
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Is equality of processes stable to multiplication with an independent process.

Assume that all processes to be considered are regular (say cadlag). Assume $X^1$ and $X^2$ are stochastic processes such that $X^1_t = X^2_t$, that $Y^1$ and $Y^2$ are processes such that ...
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Integrating over random boundary

What are some correct stochastic integral notions or theories which make formal sense of the problem of "integrating a function over the boundary of random domain"?
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40 views

Conditional independence of stopping times from i.i.d. stochastic processes

My question is somewhat arbitrary but I was thinking about independence of processes and stopping times. Say that we define two processes $X,Y$ on different probability spaces ...
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1answer
23 views

What is the conditional distribution function of the moment when the first event happens when the total number of event is given?

Suppose books are getting missing from a library as a Poisson process $N(t)$ with intensity $\lambda$. Suppose we know that at the moment $t$, $N(t) = n$. What is the conditional distribution ...
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Are sojourn times independent in non homogeneous poisson process?

We know that in homogeneous Poisson process, sojourn times are independent. But are they still independent if $\lambda (t)$ varies? If so, what is the proof? Thanks the problem I have now is that ...
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Information in Filtrations

Is the “information” kept track of by filtrations the same as information-theoretic “information”? If not, is there some way the two concepts can be reconciled?
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Survival probability of a biased random walker

A random walker moves to $+1$ with probability $p$ and moves to $-1$ with probability $q=1-p$. If he starts at point $m$, what is the probability that he doesn't hit the point zero after $k$ steps, ...
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Lévy-process property

I get a problem that comes up in the construction of the Lévy-Itõ decomposition. For a Lévy-process $X$ there is a independently scattered poisson random measure $N$, such that for each t, and for ...
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Intuition of Doob-Meyer decomposition ( case of totally inaccessible jumps)

I try to understand Theorem 10 on page 107 of Protter's Stochastic integration and differential equations. The proof is really long, and for now, I just want to get an intuition. Here is the theorem: ...
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A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that ...
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Ito's lemma in infinite dimensional spaces

I'm trying to use the ito's lemma in infinite dimensional spaces applicatte to $F(X)=\Vert AX\Vert^{2}$, where $A$ is a linear map. But i I have trouble calculating the integral $\int_{0}^{t}\langle ...
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1answer
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Sum Before Crossing Value x

I'm having trouble answering the following question: Let $X_1$, $X_2$, .... be uniform [0,1] iid RVs. Define: $t(x) = min(n>0:X_1+X_2+....X_n > x)$. Find $P(t(x)>2).$ To get an idea of ...
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Stochastics Process [closed]

There are two transatlantic cables each of which can handle one telegraph message at a time. The time-to-breakdown for each has the same exponential distribution with parameter λ. The time to repair ...
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$\mathcal{P}$ stochastic matrix. If there is $k > 0$ st $\mathcal{P}^k(j, i) > 0$, then there is $r \leq (n-1)$ st $\mathcal{P}^r(j, i) > 0$

Let $\mathcal{P}$ be stochastic matrix of order n. If there is $k > 0$ such that $\mathcal{P}^k(j, i) > 0$, then there is $r \leq (n-1)$ such that $\mathcal{P}^r(j, i) > 0$. My attempt: ...
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1answer
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Maximum process of Brownian motion

Consider the linear standard Brownian motion $(B_t)_{t \ge 0}$. We define the maximum process $(M_t)_{t \ge 0}$ of $(B_t)_{t \ge 0}$ to be such that $M_t = \max_{0\le s \le t} B_s$. Prove that the ...
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1answer
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Existence and uniqueness of SDE, is the independence requirement needed?

In Bernt Øksendals Stochastic differential equations he has this theorem in chapter 5: $\\\\\\$ However, in the proof I can not see where he uses the independence condition I marked in red. Do you ...
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Counterexample for “Filtration of stopping time equals filtration generated by stopped process”

I am working in a discrete setting. Consider any stochastic process $(X_n)_{n\in\mathbb N}$ with its natural filtration $(\mathcal F_n)_{n\in\mathbb N}$ and a stopping time $\tau$. We know that ...
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Verifying data came from a Wiener Process

From the Wiki article a Wiener Process has the properties that $$E[W_t] = 0$$ $$Var[W_t] = t$$ According to A Standard Wiener Process the Wiener Process is given by: $$W(t) - W(s) \tilde{} ...
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Version of the CLT

It is well known, that for a sequence of i.i.d. rv. $X_i$ with $E[X_i]=\mu$ and $Var[X_i]=\sigma^{2}$ that $$ ...
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Question about “Stochastic Analysis on Manifolds”

After Definition 2.3.1 Hsu says that if $M$ is a closed submanifold of $\mathbb{R}^N$ then a semimartingale $X$ on $M\subseteq\mathbb{R}^N$ should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ ...
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Conditional Distributions vs. Stochastic Processes

Is the concept of a version of a stochastic process related to the concept of a version of a conditional distribution? And is a regular version of a stochastic process somehow the same thing as the ...
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1answer
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Almost sure convergence of the inverse

If a sequence of non-negative random variables $X_1, X_2, \dots$ converges almost surely to a random variable $X$, that is $X_n \xrightarrow{a.s} X$ or equivalently ...
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Convergence in distribution for two random variables

If $\lim\limits_{n \to \infty} P(X_n\leq T)=P(X\leq T)$ and $\lim\limits_{n \to \infty} P(Y_n\leq T)=P(Y\leq T)$, where $X_1, X_2,\cdots$ and $Y_1, Y_2,\cdots$ are two sequences of random ...
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$X$ is a Right-Continuous process iff $\mathcal{F}^X$ filtration is RC?

I have a doubt on this assertion : $X$ is a right-continuous adapted process $\iff$ $\mathcal{F}^X$ is right-continuous ? I have mainly a doubt on this direction $\Leftarrow$, I do not find a mean ...
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31 views

Wiener process and stochastic int

Let $h:[0,1] \rightarrow \left\{-1,1 \right\}$. How to show that $X_t=(\int_0^th(s)dW_s)_{t \in [0,1]}$ is a Wiener process? I know from the lecture that for every $h$ process $\int h \ dW_s$ is ...
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Stochastic Integral with respect to Compensated Poisson Process

Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral ...